### Existence and concentration of local mountain passes for a nonlinear elliptic field equation in the semi-classical limit

DOI: http://dx.doi.org/10.12775/TMNA.2001.015

#### Abstract

In this paper we are concerned with the problem of finding solutions for

the following nonlinear field equation

$$

-\Delta u + V(hx)u-\Delta_{p}u+ W'(u)=0,

$$

where $u:\mathbb R^{N}\rightarrow \mathbb R^{N+1}$, $N\geq3$, $p> N$ and $h> 0$.

We assume that the potential $V$ is positive and $W$ is an appropriate

singular function. In particular we deal with the existence of solutions

obtained as critical (not minimum) points for the associated energy functional

when $h$ is small enough. Such solutions will eventually exhibit some notable

behaviour as $h\rightarrow 0^{+}$. The proof of our results is variational

and consists in the introduction of a modified (penalized) energy functional

for which mountain pass solutions are studied and soon after are proved

to solve our equation for $h$ sufficiently small. This idea is in the spirit

of that used in M. Del Pino and P. Felmer

[< i> Local mountain passes for semilinear elliptic problems

in unbounded domains< /i> , Calc. Var. Partial Differential Equations < b> 4< /b> (1996), 121–137],

[< i> Semi-classical states for nonlinear Schrödinger equations< /i> , J. Funct. Anal. < b> 149< /b>

(1997), 245–265]

and

[< i> Multi-peak bound states for nonlinear Schrödinger equations< /i> , Ann. Inst. H. Poincaré

Anal. Non Linéaire < b> 15< /b> (1998), 127–149], where "local

mountain passes" are found in certain nonlinear Schrödinger equations.

the following nonlinear field equation

$$

-\Delta u + V(hx)u-\Delta_{p}u+ W'(u)=0,

$$

where $u:\mathbb R^{N}\rightarrow \mathbb R^{N+1}$, $N\geq3$, $p> N$ and $h> 0$.

We assume that the potential $V$ is positive and $W$ is an appropriate

singular function. In particular we deal with the existence of solutions

obtained as critical (not minimum) points for the associated energy functional

when $h$ is small enough. Such solutions will eventually exhibit some notable

behaviour as $h\rightarrow 0^{+}$. The proof of our results is variational

and consists in the introduction of a modified (penalized) energy functional

for which mountain pass solutions are studied and soon after are proved

to solve our equation for $h$ sufficiently small. This idea is in the spirit

of that used in M. Del Pino and P. Felmer

[< i> Local mountain passes for semilinear elliptic problems

in unbounded domains< /i> , Calc. Var. Partial Differential Equations < b> 4< /b> (1996), 121–137],

[< i> Semi-classical states for nonlinear Schrödinger equations< /i> , J. Funct. Anal. < b> 149< /b>

(1997), 245–265]

and

[< i> Multi-peak bound states for nonlinear Schrödinger equations< /i> , Ann. Inst. H. Poincaré

Anal. Non Linéaire < b> 15< /b> (1998), 127–149], where "local

mountain passes" are found in certain nonlinear Schrödinger equations.

#### Keywords

Nonlinear Schrödinger equation; existence. concentration; topological charge

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